Download 2. Patterns, Functions, and Algebraic Structures

2. Patterns, Functions, and Algebraic Structures
Pattern sense gives students a lens with which to understand trends and commonalities. Being a student of mathematics
involves recognizing and representing mathematical relationships and analyzing change. Students learn that the structures
of algebra allow complex ideas to be expressed succinctly.
Prepared Graduates
The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who
complete the Colorado education system must have to ensure success in a postsecondary and workforce setting.
Prepared Graduate Competencies in the 2. Patterns, Functions, and Algebraic
Structures Standard are:

Are fluent with basic numerical and symbolic facts and algorithms, and are able to select
and use appropriate (mental math, paper and pencil, and technology) methods based on
an understanding of their efficiency, precision, and transparency

Understand that equivalence is a foundation of mathematics represented in numbers,
shapes, measures, expressions, and equations

Make sound predictions and generalizations based on patterns and relationships that arise
from numbers, shapes, symbols, and data

Make claims about relationships among numbers, shapes, symbols, and data and defend
those claims by relying on the properties that are the structure of mathematics

Use critical thinking to recognize problematic aspects of situations, create mathematical
models, and present and defend solutions
Colorado Department of Education: High School Mathematics
Revised: December 2010
Page 1 of 7
Content Area: Mathematics
Standard: 2. Patterns, Functions, and Algebraic Structures
Prepared Graduates:
 Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and
data
Grade Level Expectation: High School
Concepts and skills students master:
1. Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using
tables
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
a. Formulate the concept of a function and use function notation. (CCSS: F-IF)
i. Explain that a function is a correspondence from one set (called the domain)
to another set (called the range) that assigns to each element of the domain
exactly one element of the range.1 (CCSS: F-IF.1)
ii. Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context. (CCSS:
F-IF.2)
iii. Demonstrate that sequences are functions,2 sometimes defined recursively,
whose domain is a subset of the integers. (CCSS: F-IF.3)
b. Interpret functions that arise in applications in terms of the context. (CCSS: F-IF)
i. For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs
showing key features3 given a verbal description of the relationship. ★ (CCSS:
F-IF.4)
ii. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes.4 ★ (CCSS: F-IF.5)
iii. Calculate and interpret the average rate of change 5 of a function over a
specified interval. Estimate the rate of change from a graph.★ (CCSS: F-IF.6)
c. Analyze functions using different representations. (CCSS: F-IF)
i.
Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases. ★
(CCSS: F-IF.7)
ii.
Graph linear and quadratic functions and show intercepts, maxima, and
minima. (CCSS: F-IF.7a)
iii.
Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions. (CCSS: F-IF.7b)
iv.
Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior. (CCSS: F-IF.7c)
v.
Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude. (CCSS: F-IF.7e)
vi.
Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function. (CCSS: F-IF.8)
Colorado Department of Education: High School Mathematics
Inquiry Questions:
1. Why are relations and functions represented in multiple
ways?
2. How can a table, graph, and function notation be used to
explain how one function family is different from and/or
similar to another?
3. What is an inverse?
4. How is “inverse function” most likely related to addition and
subtraction being inverse operations and to multiplication
and division being inverse operations?
5. How are patterns and functions similar and different?
6. How could you visualize a function with four variables, such
as x
7.
8.
2
 y 2  z 2  w2  1 ?
Why couldn’t people build skyscrapers without using
functions?
How do symbolic transformations affect an equation,
inequality, or expression?
Relevance and Application:
1. Knowledge of how to interpret rate of change of a function
allows investigation of rate of return and time on the value
of investments. (PFL)
2. Comprehension of rate of change of a function is important
preparation for the study of calculus.
3. The ability to analyze a function for the intercepts,
asymptotes, domain, range, and local and global behavior
provides insights into the situations modeled by the
function. For example, epidemiologists could compare the
rate of flu infection among people who received flu shots to
the rate of flu infection among people who did not receive a
flu shot to gain insight into the effectiveness of the flu shot.
4. The exploration of multiple representations of functions
develops a deeper understanding of the relationship
Revised: December 2010
Page 2 of 7
1. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context. (CCSS: F-IF.8a)
2. Use the properties of exponents to interpret expressions for exponential
functions.6 (CCSS: F-IF.8b)
3. Compare properties of two functions each represented in a different way7
(algebraically, graphically, numerically in tables, or by verbal
descriptions). (CCSS: F-IF.9)
d. Build a function that models a relationship between two quantities. (CCSS: F-BF)
i. Write a function that describes a relationship between two quantities.★ (CCSS:
F-BF.1)
1. Determine an explicit expression, a recursive process, or steps for
calculation from a context. (CCSS: F-BF.1a)
2. Combine standard function types using arithmetic operations.8 (CCSS: FBF.1b)
ii. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.★ (CCSS: F-BF.2)
e. Build new functions from existing functions. (CCSS: F-BF)
i. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and
f(x + k) for specific values of k,9 and find the value of k given the graphs.10
(CCSS: F-BF.3)
ii. Experiment with cases and illustrate an explanation of the effects on the graph
using technology.
iii. Find inverse functions.11 (CCSS: F-BF.4)
f. Extend the domain of trigonometric functions using the unit circle. (CCSS: F-TF)
i. Use radian measure of an angle as the length of the arc on the unit circle
subtended by the angle. (CCSS: F-TF.1)
ii. Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of
angles traversed counterclockwise around the unit circle. (CCSS: F-TF.2)
*Indicates a part of the standard connected to the mathematical practice of Modeling
Colorado Department of Education: High School Mathematics
between the variables in the function.
5. The understanding of the relationship between variables in a
function allows people to use functions to model
relationships in the real world such as compound interest,
population growth and decay, projectile motion, or payment
plans.
6. Comprehension of slope, intercepts, and common forms of
linear equations allows easy retrieval of information from
linear models such as rate of growth or decrease, an initial
charge for services, speed of an object, or the beginning
balance of an account.
7. Understanding sequences is important preparation for
calculus. Sequences can be used to represent functions
including e
x
2
, e x , sin x, and cos x .
Nature of Mathematics:
1. Mathematicians use multiple representations of functions
to explore the properties of functions and the properties
of families of functions.
2. Mathematicians model with mathematics. (MP)
3. Mathematicians use appropriate tools strategically. (MP)
4. Mathematicians look for and make use of structure. (MP)
Revised: December 2010
Page 3 of 7
Content Area: Mathematics
Standard: 2. Patterns, Functions, and Algebraic Structures
Prepared Graduates:
 Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend
solutions
Grade Level Expectation: High School
Concepts and skills students master:
2. Quantitative relationships in the real world can be modeled and solved using functions
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
a. Construct and compare linear, quadratic, and exponential models and solve
problems. (CCSS: F-LE)
i. Distinguish between situations that can be modeled with linear functions
and with exponential functions. (CCSS: F-LE.1)
1. Prove that linear functions grow by equal differences over equal
intervals, and that exponential functions grow by equal factors over
equal intervals. (CCSS: F-LE.1a)
2. Identify situations in which one quantity changes at a constant rate
per unit interval relative to another. (CCSS: F-LE.1b)
3. Identify situations in which a quantity grows or decays by a constant
percent rate per unit interval relative to another. (CCSS: F-LE.1c)
ii. Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
input-output pairs.12 (CCSS: F-LE.2)
iii. Use graphs and tables to describe that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more
generally) as a polynomial function. (CCSS: F-LE.3)
iv. For exponential models, express as a logarithm the solution to abct = d
where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the
logarithm using technology. (CCSS: F-LE.4)
b. Interpret expressions for function in terms of the situation they model. (CCSS:
F-LE)
i. Interpret the parameters in a linear or exponential function in terms of a
context. (CCSS: F-LE.5)
c. Model periodic phenomena with trigonometric functions. (CCSS: F-TF)
i. Choose the trigonometric functions to model periodic phenomena with
specified amplitude, frequency, and midline. ★ (CCSS: F-TF.5)
d. Model personal financial situations
i. Analyze* the impact of interest rates on a personal financial plan (PFL)
ii. Evaluate* the costs and benefits of credit (PFL)
iii. Analyze various lending sources, services, and financial institutions (PFL)
*Indicates a part of the standard connected to the mathematical practice of Modeling.
Colorado Department of Education: High School Mathematics
Inquiry Questions:
1. Why do we classify functions?
2. What phenomena can be modeled with particular functions?
3. Which financial applications can be modeled with exponential
functions? Linear functions? (PFL)
4. What elementary function or functions best represent a given
scatter plot of two-variable data?
5. How much would today’s purchase cost tomorrow? (PFL)
Relevance and Application:
1. The understanding of the qualitative behavior of functions allows
interpretation of the qualitative behavior of systems modeled by
functions such as time-distance, population growth, decay, heat
transfer, and temperature of the ocean versus depth.
2. The knowledge of how functions model real-world phenomena
allows exploration and improved understanding of complex
systems such as how population growth may affect the
environment , how interest rates or inflation affect a personal
budget, how stopping distance is related to reaction time and
velocity, and how volume and temperature of a gas are related.
3. Biologists use polynomial curves to model the shapes of jaw
bone fossils. They analyze the polynomials to find potential
evolutionary relationships among the species.
4. Physicists use basic linear and quadratic functions to model the
motion of projectiles.
Nature of Mathematics:
1. Mathematicians use their knowledge of functions to create
accurate models of complex systems.
2. Mathematicians use models to better understand systems and
make predictions about future systemic behavior.
3. Mathematicians reason abstractly and quantitatively. (MP)
4. Mathematicians construct viable arguments and critique the
reasoning of others. (MP)
5. Mathematicians model with mathematics. (MP)
Revised: December 2010
Page 4 of 7
Content Area: Mathematics
Standard: 2. Patterns, Functions, and Algebraic Structures
Prepared Graduates:
 Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations
Grade Level Expectation: High School
Concepts and skills students master:
3. Expressions can be represented in multiple, equivalent forms
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
Inquiry Questions:
a. Interpret the structure of expressions.(CCSS: A-SSE)
1. When is it appropriate to simplify expressions?
i. Interpret expressions that represent a quantity in terms of its context.★
2. The ancient Greeks multiplied binomials and found the roots of
(CCSS: A-SSE.1)
quadratic equations without algebraic notation. How can this be
1. Interpret parts of an expression, such as terms, factors, and
done?
coefficients. (CCSS: A-SSE.1a)
Interpret complicated expressions by viewing one or more of their parts
as a single entity.13 (CCSS: A-SSE.1b)
ii. Use the structure of an expression to identify ways to rewrite it.14 (CCSS: ASSE.2)
Write expressions in equivalent forms to solve problems. (CCSS: A-SSE)
i. Choose and produce an equivalent form of an expression to reveal and explain
properties of the quantity represented by the expression.★ (CCSS: A-SSE.3)
1. Factor a quadratic expression to reveal the zeros of the function it
defines. (CCSS: A-SSE.3a)
2. Complete the square in a quadratic expression to reveal the maximum
or minimum value of the function it defines. (CCSS: A-SSE.3b)
3. Use the properties of exponents to transform expressions for
exponential functions.15 (CCSS: A-SSE.3c)
ii. Derive the formula for the sum of a finite geometric series (when the common
ratio is not 1), and use the formula to solve problems.16★ (CCSS: A-SSE.4)
Perform arithmetic operations on polynomials. (CCSS: A-APR)
i. Explain that polynomials form a system analogous to the integers, namely,
they are closed under the operations of addition, subtraction, and
multiplication; add, subtract, and multiply polynomials. (CCSS: A-APR.1)
Understand the relationship between zeros and factors of polynomials. (CCSS: AAPR)
i. State and apply the Remainder Theorem.17 (CCSS: A-APR.2)
ii. Identify zeros of polynomials when suitable factorizations are available, and
use the zeros to construct a rough graph of the function defined by the
polynomial. (CCSS: A-APR.3)
Use polynomial identities to solve problems. (CCSS: A-APR)
i. Prove polynomial identities18 and use them to describe numerical relationships.
(CCSS: A-APR.4)
Rewrite rational expressions. (CCSS: A-APR)
Rewrite simple rational expressions in different forms.19 (CCSS: A-APR.6)
2.
b.
c.
d.
e.
f.
g.
Relevance and Application:
1. The simplification of algebraic expressions and solving equations
are tools used to solve problems in science. Scientists represent
relationships between variables by developing a formula and using
values obtained from experimental measurements and algebraic
manipulation to determine values of quantities that are difficult or
impossible to measure directly such as acceleration due to gravity,
speed of light, and mass of the earth.
2. The manipulation of expressions and solving formulas are
techniques used to solve problems in geometry such as finding the
area of a circle, determining the volume of a sphere, calculating the
surface area of a prism, and applying the Pythagorean Theorem.
Nature of Mathematics:
1. Mathematicians abstract a problem by representing it as an
equation. They travel between the concrete problem and the
abstraction to gain insights and find solutions.
2. Mathematicians construct viable arguments and critique the
reasoning of others. (MP)
3. Mathematicians model with mathematics. (MP)
4. Mathematicians look for and express regularity in repeated
reasoning. (MP)
*Indicates a part of the standard connected to the mathematical practice of Modeling
Colorado Department of Education: High School Mathematics
Revised: December 2010
Page 5 of 7
Content Area: Mathematics
Standard: 2. Patterns, Functions, and Algebraic Structures
Prepared Graduates:
 Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper
and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency
Grade Level Expectation: High School
Concepts and skills students master:
4. Solutions to equations, inequalities and systems of equations are found using a variety of tools
Evidence Outcomes
Students can:
a. Create equations that describe numbers or relationships. (CCSS: A-CED)
i. Create equations and inequalities20 in one variable and use them to solve problems. (CCSS: A-CED.1)
ii. Create equations in two or more variables to represent relationships between quantities and graph
equations on coordinate axes with labels and scales. (CCSS: A-CED.2)
iii. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and
interpret solutions as viable or nonviable options in a modeling context.21 (CCSS: A-CED.3)
iv. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.22
(CCSS: A-CED.4)
b. Understand solving equations as a process of reasoning and explain the reasoning. (CCSS: A-REI)
i. Explain each step in solving a simple equation as following from the equality of numbers asserted at the
previous step, starting from the assumption that the original equation has a solution. (CCSS: A-REI.1)
ii. Solve simple rational and radical equations in one variable, and give examples showing how extraneous
solutions may arise. (CCSS: A-REI.2)
c. Solve equations and inequalities in one variable. (CCSS: A-REI)
i. Solve linear equations and inequalities in one variable, including equations with coefficients represented by
letters. (CCSS: A-REI.3)
ii. Solve quadratic equations in one variable. (CCSS: A-REI.4)
1. Use the method of completing the square to transform any quadratic equation in x into an equation of
the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. (CCSS:
A-REI.4a)
2. Solve quadratic equations23 by inspection, taking square roots, completing the square, the quadratic
formula and factoring, as appropriate to the initial form of the equation. (CCSS: A-REI.4b)
3. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b. (CCSS: A-REI.4b)
d. Solve systems of equations. (CCSS: A-REI)
i. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that
equation and a multiple of the other produces a system with the same solutions. (CCSS: A-REI.5)
ii. Solve systems of linear equations exactly and approximately,24 focusing on pairs of linear equations in two
variables. (CCSS: A-REI.6)
iii. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically
and graphically.25 (CCSS: A-REI.7)
e. Represent and solve equations and inequalities graphically. (CCSS: A-REI)
i. Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate
plane, often forming a curve.26 (CCSS: A-REI.10)
ii. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x);27 find the solutions approximately.28★ (CCSS: AREI.11)
iii. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the
case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as
the intersection of the corresponding half-planes. (CCSS: A-REI.12)
*Indicates a part of the standard connected to the mathematical practice of Modeling
Colorado Department of Education: High School Mathematics
Revised: December 2010
21st Century Skills and Readiness Competencies
Inquiry Questions:
1. What are some similarities in solving all types of
equations?
2. Why do different types of equations require
different types of solution processes?
3. Can computers solve algebraic problems that
people cannot solve? Why?
4. How are order of operations and operational
relationships important when solving
multivariable equations?
Relevance and Application:
1. Linear programming allows representation of the
constraints in a real-world situation identification
of a feasible region and determination of the
maximum or minimum value such as to optimize
profit, or to minimize expense.
2. Effective use of graphing technology helps to find
solutions to equations or systems of equations.
Nature of Mathematics:
1. Mathematics involves visualization.
2. Mathematicians use tools to create visual
representations of problems and ideas that reveal
relationships and meaning.
3. Mathematicians construct viable arguments and
critique the reasoning of others. (MP)
4. Mathematicians use appropriate tools
strategically. (MP)
Page 6 of 7
Standard: 2. Patterns, Functions, and Algebraic Structures
High School
If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph
of the equation y = f(x). (CCSS: F-IF.1)
2
For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. (CCSS: F-IF.3)
3
Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and
minimums; symmetries; end behavior; and periodicity. (CCSS: F-IF.4)
4
For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers
would be an appropriate domain for the function. (CCSS: F-IF.5)
5
presented symbolically or as a table. (CCSS: F-IF.6)
6
For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10,. (CCSS: F-IF.8b)
7
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (CCSS: FIF.9)
8
For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and
relate these functions to the model. (CCSS: F-BF.1b)
9
both positive and negative. (CCSS: F-BF.3)
10
Include recognizing even and odd functions from their graphs and algebraic expressions for them. (CCSS: F-BF.3)
11
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. (CCSS: F-BF.4a)
12
include reading these from a table. (CCSS: F-LE.2)
13
For example, interpret P(1+r)n as the product of P and a factor not depending on P. (CCSS: A-SSE.1b)
14
For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x 2 – y2)(x2 + y2). (CCSS: ASSE.2)
15
For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if
the annual rate is 15%. (CCSS: A-SSE.3c)
16
For example, calculate mortgage payments. (CCSS: A-SSE.4)
17
For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
(CCSS: A-APR.2)
18
For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. (CCSS: A-APR.4)
19
write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x),
using inspection, long division, or, for the more complicated examples, a computer algebra system. (CCSS: A-APR.6)
20
Include equations arising from linear and quadratic functions, and simple rational and exponential functions. (CCSS: A-CED.1)
21
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. (CCSS: A-CED.3)
22
For example, rearrange Ohm’s law V = IR to highlight resistance R. (CCSS: A-CED.4)
23
e.g., for x2 = 49. (CCSS: A-REI.4b)
24
e.g., with graphs. (CCSS: A-REI.6)
25
For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. (CCSS: A-REI.7)
26
which could be a line. (CCSS: A-REI.10)
27
Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (CCSS: AREI.11)
28
e.g., using technology to graph the functions, make tables of values, or find successive approximations. (CCSS: A-REI.11)
1
Colorado Department of Education: High School Mathematics
Revised: December 2010
Page 7 of 7